L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.212 − 0.977i)3-s + (0.841 − 0.540i)4-s + (0.755 + 0.654i)5-s + (0.479 + 0.877i)6-s + (0.997 − 0.0713i)7-s + (−0.654 + 0.755i)8-s + (−0.909 + 0.415i)9-s + (−0.909 − 0.415i)10-s + (0.654 + 0.755i)11-s + (−0.707 − 0.707i)12-s + (−0.977 + 0.212i)13-s + (−0.936 + 0.349i)14-s + (0.479 − 0.877i)15-s + (0.415 − 0.909i)16-s + (−0.281 + 0.959i)17-s + ⋯ |
L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.212 − 0.977i)3-s + (0.841 − 0.540i)4-s + (0.755 + 0.654i)5-s + (0.479 + 0.877i)6-s + (0.997 − 0.0713i)7-s + (−0.654 + 0.755i)8-s + (−0.909 + 0.415i)9-s + (−0.909 − 0.415i)10-s + (0.654 + 0.755i)11-s + (−0.707 − 0.707i)12-s + (−0.977 + 0.212i)13-s + (−0.936 + 0.349i)14-s + (0.479 − 0.877i)15-s + (0.415 − 0.909i)16-s + (−0.281 + 0.959i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.201367450 + 0.2589546009i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.201367450 + 0.2589546009i\) |
\(L(1)\) |
\(\approx\) |
\(0.8649796237 + 0.04206705749i\) |
\(L(1)\) |
\(\approx\) |
\(0.8649796237 + 0.04206705749i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 89 | \( 1 \) |
good | 2 | \( 1 + (-0.959 + 0.281i)T \) |
| 3 | \( 1 + (-0.212 - 0.977i)T \) |
| 5 | \( 1 + (0.755 + 0.654i)T \) |
| 7 | \( 1 + (0.997 - 0.0713i)T \) |
| 11 | \( 1 + (0.654 + 0.755i)T \) |
| 13 | \( 1 + (-0.977 + 0.212i)T \) |
| 17 | \( 1 + (-0.281 + 0.959i)T \) |
| 19 | \( 1 + (0.936 + 0.349i)T \) |
| 23 | \( 1 + (0.349 - 0.936i)T \) |
| 29 | \( 1 + (0.0713 + 0.997i)T \) |
| 31 | \( 1 + (0.349 + 0.936i)T \) |
| 37 | \( 1 + (0.707 - 0.707i)T \) |
| 41 | \( 1 + (0.977 + 0.212i)T \) |
| 43 | \( 1 + (0.0713 - 0.997i)T \) |
| 47 | \( 1 + (-0.540 - 0.841i)T \) |
| 53 | \( 1 + (0.540 - 0.841i)T \) |
| 59 | \( 1 + (0.212 - 0.977i)T \) |
| 61 | \( 1 + (-0.800 + 0.599i)T \) |
| 67 | \( 1 + (0.841 + 0.540i)T \) |
| 71 | \( 1 + (-0.755 + 0.654i)T \) |
| 73 | \( 1 + (-0.415 + 0.909i)T \) |
| 79 | \( 1 + (0.909 + 0.415i)T \) |
| 83 | \( 1 + (-0.479 - 0.877i)T \) |
| 97 | \( 1 + (-0.654 + 0.755i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.71451921608896378905548071013, −28.995371832275926583350770485894, −27.87099067111954279053477400983, −27.23448117891687051400875250199, −26.36319571846278759948223968752, −24.939590093958396994938831522680, −24.350624157896617173949961726058, −22.25744056137666303669987329341, −21.386981112251812869393307868657, −20.63889400482252870602434299413, −19.67042879650907888230273542000, −17.95607331460038727279438417353, −17.23605947719368919240322548129, −16.415529773463466785635402772512, −15.18018931083309481207465092880, −13.79765411992024811406574114660, −11.888919116143199503523748570950, −11.1791373709872158986685496589, −9.70126411529674544487785124436, −9.16390837202192692012439548008, −7.82041545036125390269234642892, −5.9030493496686325056489147028, −4.61815591549117529872395308431, −2.71015848476210799892755930080, −0.89979089810158511031411917582,
1.39498532730516997691634433302, 2.35783504203768932544779590336, 5.31970480453072962182874402865, 6.66961610596499015510774545774, 7.416902030527137404826765161107, 8.74759037294301315834189644010, 10.16804135097775367120922573754, 11.2713213658337787359252675929, 12.41611974706032615400330336113, 14.28505128769356726835984161206, 14.73171622038616386427962068397, 16.78484142736907243580569548300, 17.633435042458801747369403244518, 18.14413023957308429477707393105, 19.32979491418074767835100658814, 20.339813483850745394888936544778, 21.821871802833340133206492141734, 23.18167068846686080522998267252, 24.48029525962430029757328820522, 24.919710253578401108322582501471, 26.09088595862583153563864652019, 27.10030808476488001485160299105, 28.36552645897068162703755374359, 29.18026692887440928358675420154, 30.19376148307958166916548724409